Bounty - When ~ enters the battlefield, place a bounty on target Creature. When that creature dies - effect.
or
Bounty - (You may pair this creature with another unpaired creature an opponent controls when either enters the battlefield. They remain paired until one leaves the battlefield.)

I like the first Bounty.
I imagine it playing out quite like a real outlaw scenario in which the hunted is trying to either kill is hunter first or lay low long enough that Life will deal with his hunter. Very cool.

I also like the first Bounty a lot. To avoid memory issues I would want to change it to:
Bounty - When ~ enters the battlefield, place a bounty on target Creature. When a creature with a bounty counter dies gain life equal to its toughness.

That way it gets less confusing if there are multiple creatures with Bounty counters on them and you're trying to remember who marked which card. Plus, it encourages you to play Bounty cards together— A deck full of Sheriffs sounds pretty fun.

The basic idea seems cool, but there are some weird flavor tics in this implementation. You can put bounties on your own creatures. In fact, if you play a Sheriff on an empty board, he's forced to put a bounty on himself! Also, you get the reward for the creature dying no matter who killed it, and multiple people can collect the same reward.